In our recent work [1]}, we showed that in one dimension and in the context of multigrid, an element-wise Vanka smoother is equivalent to the scaled mass operator obtained from the linear finite element method, and in two dimensions, the
element-wise Vanka smoother is equivalent to the scaled mass operator discretized by bilinear finite element method plus a scaled identity operator. While the context of that work is different, this has motivated us to ask whether the mass matrix obtained from the finite element method can be utilized as a preconditioner for the Laplacian. Here, we mean that preconditioning would amount to multiplying by the mass matrix; no inversion is involved. Such a possibility seems attractive given the ease of multiplying the Laplacian by the mass matrix, which is sparse and well conditioned. In this short note we provide analytical and numerical evidence that using the mass matrix in this manner at least doubles the convergence speed of CG in 2D and 3D at a modest computational cost.
